Embedding 3-manifolds with circle actions in 4-space

Jonathan A. Hillman


We give constraints on the Seifert invariants of closed orientable 3-manifolds M which admit fixed-point free circle actions and embed in 4-space. In particular, the generalized Euler invariant of the orbit fibration is determined up to sign by the base orbifold B unless H1(M) is torsion free, in which case it can take at most one nonzero value (up to sign). No such manifold with base B=S2(a1,…,ar) with r odd and Euler invariant 0 embeds in 4-space. [This is an extensive revision of a 1998 Research Report.]

Keywords: embedding. Euler invariant. linking pairing. Seifert bundle.

AMS Subject Classification: Primary 57N10.

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Friday, June 13, 2008